Soft robots show an advantage when conducting tasks in complex environments due to their enormous flexibility and adaptability. However, soft robots suffer interactions and nonlinear deformation when interacting with soft and fluid materials. The reason behind is the free boundary interactions, which refers to undetermined contact between soft materials, specifically containing nonlinear deformation in air and nonlinear interactions in fluid for soft robot simulation. Therefore, we propose a new approach using material point method (MPM), which can solve the free boundary interactions problem, to simulate soft robots under such environments. The proposed approach can autonomously predict the flexible and versatile behaviors of soft robots. Our approach entails incorporating automatic differentiation into the algorithm of MPM to simplify the computation and implement an efficient implicit time integration algorithm. We perform two groups of experiments with an ordinary pneumatic soft finger in different free boundary interactions. The results indicate that it is possible to simulate soft robots with nonlinear interactions and deformation, and such environmental effects on soft robots can be restored.

Naturally deformed biotite in contact-metamorphosed slates affected by a shear zone of the Southern Iberian Massif near Jaén (SE Spain) were studied by X-ray diffraction, optical microscopy, scanning electron microscopy, electron probe microanalysis and high-resolution transmission and analytical electron microscopy. Biotite is found in the contact metamorphism aureole produced by the intrusion of a granodioritic stock, but shear strain caused its deformation. The southern part of the shear band is strongly deformed, containing thick clay gouge zones. The northern part is less deformed and develops weaker planar-linear fabrics. X-ray diffraction data reveal the predominance of the 2M1 biotite in the undeformed samples whereas the 1M polytype is predominant in the sheared samples. Chemical data and electron images of the biotite from unsheared slates do not show the presence of intercalated phases. This biotite is almost defect-free and electron diffraction patterns have sharp reflections indicating a two-layer polytype (probably 2M1). Back-scattered electron images from the deformed biotite in the moderate deformation part of the shear zone do not reveal intergrown minerals, but the electron microprobe analyses show some Fe- and Mg-enriched compositions. Transmission electron microscopy indicates that disordered polytype packets are predominant (probably 1Md). Their electron diffraction patterns have diffuse streaking along c*. These packets have high dislocation densities, microcavities with ∼5 Å latticefringe regions (probably brucite-like sheets) and interlayering of chlorite-berthierine. Kaolinized biotite can be observed in the clay gouges from the strongly deformed south part of the shear zone. The degree of streaking, as an indication of the intensity of deformation, revealed that the disordered polytype packets are more deformed than the two-layer polytype packets. The microcavities of the disordered polytype packets could act as potential channels for transport of fluids during the shearing stage and serve as sites for chloritization of biotite, producing chlorite-berthierine domains within biotite. Berthierine is an intermediate metastable phase replaced by chlorite with along-layer transitions.

The purpose of this study was to investigate the structural and chemical modifications of phyllosilicates that occur under natural conditions, using the progressive deformation of chlorite (sudoite) present in quartz-rich veins from the Internal Zone of the Rif range (Morocco) as the model system. Signs of chlorite deformation include kinks, chevron-like folds, and fractures. The samples also contain later, undeformed grains, which sealed the fractures or grew with (001) perpendicular to the compressive stress. Deformation-induced structural changes consist mainly of basal cleavages associated with ordered replacement of brucite sheets by hydrated layers, thus leading to irregular microdomains of mixed-layer chlorite-vermiculite and sudoite. Such structural modifications represent a mechanism for accommodating the compressive stress. Structural changes were accompanied by minor chemical ones, which lead from di,tri-chlorite (sudoite) to phases with a more trioctahedral character (mixed-layer chlorite-vermiculite). The hydration reaction occurred throughout a topotactic replacement of the pre-existing sudoite grains. Later, undeformed grains consist of mixed-layer chlorite-vermiculite intergrown with retrogressive kaolinite and minor Fe oxide, and are interpreted as having formed through a dissolution-precipitation process, during deformation. Retrogression of sudoite probably occurred during the latest stage of exhumation, in low-temperature conditions.

A meta-porphyritic granitoid in the Makrohar Granulite Belt, Central India contains extensive myrmekite. This work evaluates the controls of fluid in relation to deformation and the formation of myrmekite all along the periphery of an alkali-feldspar megacryst. Two different myrmekite morphologies are present: (1) vermicular intergrowth of plagioclase (An38–39) and quartz (Myr1); and (2) polygonal aggregates of coarse plagioclase (An45–46) and quartz (Myr2). Petrographic features suggest that myrmekite Myr1 nucleates on alkali-feldspar and plagioclase porphyroclasts and the myrmekite front moved into the alkali feldspar by replacing it; and that myrmekite Myr2 and the secondary biotite which replaces plagioclase porphyroclasts and garnet form together. Deformation had a decisive role in forming the polygonal aggregates of Myr2, however field and microtextural features do not support any significant control of deformation during the formation of Myr1. Reaction modelling and a mass-balance calculation suggest that Ca and Na are added to, and K is removed from, the alkali feldspar during the myrmekite formation at nearly constant Si and Al. However, the secondary biotite-forming reaction, consumes K and releases Ca. Interpretation of the reaction textures in different isothermal–isobaric sections of μK2O–μCaO in the KCFASH system suggest that CaO and K2O moved in opposite directions for myrmekitisation and along their respective chemical potential gradients created between the sites of formation of myrmekite and secondary biotite. The feedback mechanism which operated between the two reaction sites was controlled by infiltration of brine-rich fluid in the meta-granitoid during a regional hydration event (550–600oC and 5–6 kbar). Volume reduction of ~10% during the formation Myr1 and Myr2 drew the brine-rich fluid towards the alkali feldspar and thus facilitated the process of myrmekite formation. Variation in the morphology of quartz in the myrmekite is attributed to the cooling of the complex.

This paper introduces a generalization of the $dd^c$-condition for complex manifolds. Like the $dd^c$-condition, it admits a diverse collection of characterizations, and is hereditary under various geometric constructions. Most notably, it is an open property with respect to small deformations. The condition is satisfied by a wide range of complex manifolds, including all compact complex surfaces, and all compact Vaisman manifolds. We show there are computable invariants of a real homotopy type which in many cases prohibit it from containing any complex manifold satisfying such $dd^c$-type conditions in low degrees. This gives rise to numerous examples of almost complex manifolds which cannot be homotopy equivalent to any of these complex manifolds.

An affine variety with an action of a semisimple group G is called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group ${\mathbb {K}^{*}}$ commuting with the G-action. We show that X is determined by the ${\mathbb {K}^{*}}$-variety $X^U$ of fixed points under a maximal unipotent subgroup $U \subset G$. Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient $X /\!\!/ G$.

If G is of type ${\mathsf {A}_n}$ ($n\geq 2$), ${\mathsf {C}_{n}}$, ${\mathsf {E}_{6}}$, ${\mathsf {E}_{7}}$, or ${\mathsf {E}_{8}}$, we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If $n \geq 5$, every smooth affine $\operatorname {\mathrm {SL}}_n$-variety of dimension $< 2n-2$ is an $\operatorname {\mathrm {SL}}_n$-vector bundle over the smooth quotient $X /\!\!/ \operatorname {\mathrm {SL}}_n$, with fiber isomorphic to the natural representation or its dual.

Soft magnetic robots have attracted tremendous interest owning to their controllability and manoeuvrability, demonstrating great prospects in a number of industrial areas. However, further explorations on the locomotion and corresponding deformation of magnetic robots with complex configurations are still challenging. In the present study, we analyse a series of soft magnetic robots with various geometric shapes under the action of the magnetic field. First, we prepared the matrix material for the robot, that is, the mixture of silicone and magnetic particles. Next, we fabricated a triangular robot whose locomotion speed and warping speed are approximately 1.5 and 9 mm/s, respectively. We then surveyed the generalised types of robots with other shapes, where the movement, grabbing, closure and flipping behaviours were fully demonstrated. The experiments show that the arching speed and grabbing speed of the cross-shaped robot are around 4.8 and 3.5 mm/s, the crawling speed of the pentagram-shaped robot is 3.5 mm/s, the pentahedron-shaped robot can finish its closure motion in 1 s and the arch-shaped robot can flip forward and backward in 0.5 s. The numerical simulation based on the finite element method has been compared with the experimental results, and they are in excellent agreement. The results are beneficial to engineer soft robots under the multi-fields, which can broaden the eyes on inventing intellectual devices and equipment.

Strain, temperature and strain rate are crucial factors governing the development of crystallographic preferred orientations (CPO) in ice. To better understand how CPO patterns change in response to these variables, we performed quantitative analyses on neutron diffraction data between 2010 and 2019, collected in situ during uniaxial compression experiments on deuterium ice. At strains >10% and temperatures <−10°C, the c-axis pattern switches from a single maximum (‘cluster’) to small circle (‘cone’), both oriented parallel to shortening. The diameter and mean width of the cone pattern decrease as strain and/or strain rate increases. Prismatic axis (a and m) patterns are characterised by great circles parallel to the pole figure margin and may be distinguishable from the patterns in ice deformed under simple shear. While strain has the main influence on the degree of preferred orientation (or CPO ‘strength’), both temperature and strain rate have minor influences, which limits the extent to which CPOs can be used to measure strain. As cluster patterns can be observed in the c-axes of ice deformed under both pure and simple shear settings, this may complicate interpretations of flow geometry in terrestrial ice unless the prismatic axis patterns are also considered.

Due to the efficient and flexible flight ability of bats, bat-like robots have become the focus of research in the field of bionic robots. Aerodynamic calculation is an important part of the research field of bat-like robot, which is the basis of the structure design and flight controller design of bat-like robot. However, due to the complex flight mechanism of bats, there is no mature theoretical method to calculate the flight aerodynamic force of bat-like robots. To solve this problem, this paper takes the membrane of a bat-like robot as the research object and analyzes in detail the effects of wing folding and unfolding and flexible deformation of the membrane on the chord length, passive torsion angle and relative velocity. Based on quasi-steady state model and blade element method, a set of aerodynamic calculation method for flexible deformed wing is established. In order to verify the effectiveness of the proposed method, the theoretical calculation results and the results of the fluid-structure interaction simulation are compared and analyzed under various working conditions. The two results are in good agreement under each working condition, and the errors are all within reasonable range, which proves the effectiveness of the method. This study can provide a theoretical basis for rational structure design and precise flight control of bat-like robot.

Understanding the distribution and geometry of faults and fractures is critical for predicting both subsurface permeability architecture and the integrity of geological fluid barriers, particularly in rocks with low primary porosity and permeability. While fracture patterns in relatively competent, weathering-resistant (therefore often well-exposed) rocks are generally well studied in outcrop, the role of mechanically weak layers in defining fracture patterns is frequently overlooked or under-represented. Here we show that rock composition, specifically clay and silicate minerals versus carbonate content, exerts a strong control on fault and fracture propagation and bed-containment within a mechanically layered, Cretaceous carbonate sequence at Canyon Lake Gorge, Texas. We find that relatively incompetent, clay-rich layers limit fault and fracture propagation, and cause bed-containment of fractures in more competent beds. In our results, no clear relationships exist between mechanical layer thickness and fracture abundance. These results are important for understanding the relative importance of composition versus bed thickness on fracture abundance in the subsurface, and for predicting fracture-controlled fluid flow pathways, seals and fracture connectivity across beds with variable compositions, thicknesses and competences.

Using crossed homomorphisms, we show that the category of weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs). This generalises and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras and to obtain new representations for generalised Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.

We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples.